The K-theory of a C*-algebra is defined in terms of equivalence classes of its projections and equivalence classes of its unitary elements — possibly after adjoining a unit and forming matrix algebras. We shall in this chapter derive the facts needed about projections and unitary elements with emphasis on the equivalence relation defined by homotopy and also — for projections — Murray–von Neumann equivalence and unitary equivalence.

Homotopy classes of unitary elements

Homotopy. Let X be a topological space. Say that two points a, b in X are homotopic in X, written a ∼hb in X, if there is a continuous function ν: [0,1] → X such that ν(0) = a and ν(1) = b. The relation ∼h is an equivalence relation on X. The continuous function ν above is called a continuous path from a to b, and it is often denoted by t ν(t) or t → νt, with or without specifying explicitly that t belongs to the interval [0,1].

Needless to say, the reference to the space X is crucial. For example, any two elements a, b in a C*-algebra A are homotopic in A. Indeed, take the continuous path t ↦ (1 − t)a + tb. But, as we shall see, two projections in A need not be homotopic in the set of all projections in A. We shall nevertheless sometimes omit the reference to the space X and just write a ∼hb instead of a ∼hb in X, when it is clear from the context in which space the homotopy should be realized.

About X-theory

K-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K-theory (for C*-algebras) is a pair of functors, called K0 and K1 that to each C*-algebra A associate two Abelian groups K0(A) and K1(A). The group K0(A) is given an ordering that (in special cases) makes it an ordered Abelian group. There are powerful machines, some of which are described in this book, making it possible to calculate the K-theory of a great many C*-algebras. K-theory contains much information about the individual C*-algebras — one can learn about the structure of a given C*-algebra by knowing its K-theory, and one can distinguish two C*-algebras from each other by distinguishing their K-theories. For certain classes of C*-algebras, K-theory is actually a complete invariant, K-theory is also a natural home for index theory.

Two applications demonstrated the importance of K-theory to C*-algebras. George Elliott showed in the early 1970s (in a work published in 1976, [18]) that AF-algebras (the so-called “approximately finite dimensional” C*-algebras; see Chapter 7 for a precise definition) are classified by their ordered K0-groups. (The K1-group of an AF-algebra is always zero.) As a consequence, all information about an AF-algebra is contained in its ordered K0-group. This result indicated the possibility of classifying a more general class of C*-algebras by their K-theory.

An Abelian group K0(A) is associated to each unital C*-algebra A. The group K0(A) arises from the Abelian semigroup (D(A), +) (defined in Chapter 2) and the Grothendieck construction (described below). We shall see that K0 is a functor from the category of unital C*-algebras to the category of Abelian groups, and some of the properties of K0 will be derived. Some examples of K0-groups can be found at the end of the chapter.

We extend K0 to a functor from the category of all C*-algebras (unital or not) in Chapter 4.

Definition of the K0-group of a unital C*-algebra

The Grothendieck construction. One can associate an Abelian group to every Abelian semigroup in a way analogous to how one obtains the integers from the natural numbers, and in much the same way as one obtains the rational numbers from the integers. We describe here how this works; the proofs of various statements along the way are deferred to the next Paragraph.

Let (S, +) be an Abelian semigroup. Define an equivalence relation ∼ on S × S by (x1, y1) ∼ (x2, y2) if there exists z in S such that x1 + y2 + z = x2 + y1 + z. That ∼ is an equivalence relation is proved in Paragraph 3.1.2.

Write G(S) for the quotient (S × S) / ∼, and let 〈 x, y 〉 denote the equiva- lence class in G(S) containing (x, y) in S × S.

An extra structure is added to the Abelian group K0(A) of a C*-algebra A by specifying a certain subset of it, called K0(A)+. The set K0(A)+ consists of all elements in K0(A) of the form [p]0, where p is a projection in P∞(A). When A is a unital, stably finite C*-algebra, then (K0(A), K0(A)+) has the pleasant structure of an ordered Abelian group. We shall for this purpose also discuss finiteness properties of C*-algebras and of projections.

The ordered K0-group of stably finite C*-algebras

An element a in a unital C*-algebra A is called left-invertible if there exists an element b in A such that ba = 1, and a is called right-invertible if ab = 1 for some b in A. If b1a = ab1 = 1, then b1 = b1ab2 = b2 and this shows that a is invertible if and only if a is both left- and right-invertible. Moreover, a is left-invertible if and only if a*a is invertible, and, similarly, a is right-invertible if and only if aa* is invertible. (See Exercise 5.1).

Definition 5.1.1. A projection p in a C*-algebra A is said to be infinite if it is equivalent to a proper subprojection of itself, i.e., if there is a projection q in A such that p ∼ q < p. If p is not infinite, then p is said to be finite.

A unital C*-algebra A is said to be finite if its unit 1A is a finite projection. Otherwise A is called infinite. If Mn(A) is finite for all positive integers n, then A is stably finite.